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After a brief exposition of the simplest class of affine theories of gravityin multidimensional space-times with symmetric connections, we consider thespherical and cylindrical reductions of these theories to two-dimensionaldilaton-vecton gravity (DVG) field theories. The distinctive feature of thesetheories is the presence of a massive/tachyonic vector field (vecton) withessentially nonlinear coupling to the dilaton gravity. In the massless limit,the classical DVG theory can be exactly solved for a rather general couplingdepending only on the field tensor and the dilaton. We show that the vectonfield can be consistently replaced by a new effectively massive scalar field(scalaron) with an unusual coupling to dilaton gravity (DSG). Then weconcentrate on considering the DVG models derived by reductions of D=3 and D=4affine theories. In particular, we introduce the most general cylindricalreductions that are often ignored. The main subject of our study is the staticsolutions with horizons. We formulate the general conditions for the existenceof the regular horizons and find the solutions of the static DVG/DSG near thehorizons in the form of locally convergent power - series expansion. For anarbitrary regular horizon, we find a local generalization of the Szekeres -Kruskal coordinates. Finally, we consider one-dimensional integrable andnonintegrable DSG theories with one scalar. We analyze simplest models havingthree or two integrals of motion, respectively, and introduce the idea of a`topological portrait' giving a unified qualitative description of static andcosmological solutions of some simple DSG models.

General properties and some solutions of generalized Einstein - Eddington affine gravity I